The index of generalised Dirac–Schrödinger operators

  • Koen van den Dungen

    SISSA, Trieste, Italy and Universität Bonn, Germany
The index of generalised Dirac–Schrödinger operators cover

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Abstract

We study the relation between spectral flow and index theory within the framework of (unbounded) K-theory. In particular, we consider a generalised notion of ‘Dirac–Schrödinger operators’, consisting of a self-adjoint elliptic first-order differential operator with a skew-adjoint ‘potential’ given by a (suitable) family of unbounded operators on an auxiliary Hilbert module. We show that such Dirac–Schrödinger operators are Fredholm, and we prove a relative index theorem for these operators (which allows cutting and pasting of the underlying manifolds). Furthermore, we show that the index of a Dirac–Schrödinger operator represents the pairing (Kasparov product) of the -theory class of the potential with the -homology class of . We prove this result without assuming that the potential is differentiable; instead, we assume that the ‘variation’ of the potential is sufficiently small near infinity. In the special case of the real line, we recover the well-known equality of the index with the spectral flow of the potential.

Cite this article

Koen van den Dungen, The index of generalised Dirac–Schrödinger operators. J. Spectr. Theory 9 (2019), no. 4, pp. 1459–1506

DOI 10.4171/JST/283